cosheaf of connected components

The display locale construction defines a fully faithful functor from cosheaves of sets on a locale $X$ to the slice category $Loc/X$ of locales over $X$.

The construction in this article describes a functor in the opposite direction that yields the inverse of the above functor once restricted to the appropriate subcategory.

Thus, the functor described here plays the same role in the equivalence between cosheaves of sets on $X$ and complete spreads over $X$ as the sheaf of sections construction plays in the equivalence between sheaves of sets on $X$ and etale locales? over $X$.

Suppose $l\colon L\to X$ is a map of locales, where $L$ is a locally connected locale. Define a precosheaf $\lambda_l$ on $X$ as follows. Send an open $U$ in $X$ to the set of connected components of $l^* U$, which is a locally connected locale because so is $L$. Send an inclusion of opens to the induced map on the sets of connected components.

(Funk, Proposition 2.1.) This precosheaf is a cosheaf.

- Jonathon Funk,
*The display locale of a cosheaf*.